Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,32)(23,33,37,43)(24,38,40,45)(25,30,26,31)(28,36,44,35)(29,41,46,39)(34,42,49,52)(47,50,51,48);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,35,27,37,26,36)(22,28,31,33,32,44,30,43)(24,34,41,50,40,49,39,48)(29,42,45,47,46,52,38,51);(7,13)(10,17)(14,19)(18,20)(21,24,26,39,27,40,25,41)(22,29,30,38,32,46,31,45)(23,34,36,48,37,49,35,50)(28,42,43,51,44,52,33,47);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,45,46,38)(34,48,49,50)(42,47,52,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,40)(29,46)(34,49)(38,45)(39,41)(42,52)(47,51)(48,50)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404908);